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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | import Mathlib
import OAI.Probability.LogConcave.JetEstimates.JetMaterial
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open scoped Classical BigOperators NNReal RealInnerProductSpace
lemma joint_mean_material {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
(hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0 < r)
(hlam : 0 < lam) (hl : (lam:ℝ)*r^2≤1/2) (h0 : 0≤ρ) (h1 : ρ<1)
(i : Fin d) (y : Point d) :
JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) (jointMean F x r i) (ρ,y)=
JetCalculus.gadj jointSpace (jointScore F x r) (fun k => jointU F x r ((lam:ℝ)*r)
(fun _ : Unit => JetCalculus.spaceBasis d k) [()] i) (ρ,y) := by
have hm := jointMean_timeSmooth hF x hr.le hl i (ρ,y) (by dsimp; linarith) h1
have ht := joint_mdir_eq_material F x r (hm.differentiableAt (by simp))
change JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) (jointMean F x r i) (ρ,y)=_ at ht
rw [ht]
change materialScalar F x r (fun t y => inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i)
(conditionalFieldMean F x r t y)) ρ y=_
rw [material_conditionalFieldMean hF x hr hlam hl h0 h1]
rw [joint_gadj_slice hF x hr.le hl h0 h1 (fun k =>
(jointU_timeSmooth hF x hr hlam hl (fun _ : Unit => JetCalculus.spaceBasis d k) [()]
(by simp) i (ρ,y) (by dsimp; linarith) h1).differentiableAt (by simp))]
rfl
lemma joint_mean_outer {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
(hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0 < r)
(hlam : 0 < lam) (hl : (lam:ℝ)*r^2≤1/2) (h0 : 0≤ρ) (h1 : ρ<1)
(i : Fin d) (y : Point d) :
JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) (jointMean F x r i) (ρ,y)=
((lam:ℝ)*r)*(TensorSum.pure TensorAtom.one).outer F x r ((lam:ℝ)*r) (fun _ => i) (ρ,y) := by
rw [joint_mean_material hF x hr hlam hl h0 h1]
have hL : (lam:ℝ)*r≠0 := ne_of_gt (mul_pos (by exact_mod_cast hlam) hr)
have he : (fun k => jointU F x r ((lam:ℝ)*r)
(fun _ : Unit => JetCalculus.spaceBasis d k) [()] i)=
fun k p => ((lam:ℝ)*r)*(TensorSum.pure TensorAtom.one).eval F x r ((lam:ℝ)*r)
(Sum.elim (fun _ => i) (fun _ => k)) p := by
funext k p
rw [TensorSum.pure_eval,TensorAtom.one_eval]
dsimp only [Sum.elim_inl,Sum.elim_inr]
field_simp
rw [he,JetCalculus.gadj_scalar_at _ _ (fun k =>
((TensorSum.pure TensorAtom.one).timeSmooth hF x hr hlam hl _ (ρ,y)
(by dsimp; linarith) h1).differentiableAt (by simp))]
rfl
theorem mean_material_representation (n : ℕ) :
∃A : TensorSum (Unit ⊕ Unit), A.weightLE (2*n) ∧
∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}, Primitive F lam →
∀(x : Point d) {r ρ : ℝ},0 < r → 0 < lam → (lam:ℝ)*r^2≤1/2 → 0≤ρ → ρ<1 →
∀(i : Fin d) (y : Point d),
JetCalculus.materialIter (1,0) jointSpace r (jointMean F x r) (n+1) (jointMean F x r i) (ρ,y)=
((lam:ℝ)*r)*A.outer F x r ((lam:ℝ)*r) (fun _ => i) (ρ,y) := by
induction n with
| zero =>
refine ⟨TensorSum.pure TensorAtom.one,TensorSum.weightLE_pure _ (by rfl),?_⟩
intro d F lam hF x r ρ hr hlam hl h0 h1 i y
exact joint_mean_outer hF x hr hlam hl h0 h1 i y
| succ n ih =>
obtain ⟨A,hA,he⟩ := ih
obtain ⟨B,hB,hAB⟩ := A.hasMaterial hA
refine ⟨B.add (A.insertOne.scale 1 Polynomial.X),?_,?_⟩
· apply TensorSum.weightLE_add
· convert hB using 1
· exact TensorSum.weightLE_mono (TensorSum.weightLE_scale
(TensorSum.weightLE_insertOne hA) _ _) (by omega)
· intro d F lam hF x r ρ hr hlam hl h0 h1 i y
have ho := A.outer_timeSmooth hF x hr hlam hl (fun _ : Unit => i)
have hs : JetCalculus.TimeSmooth (fun p => ((lam:ℝ)*r)*A.outer F x r ((lam:ℝ)*r) (fun _ => i) p) :=
fun p hp0 hp1 => contDiffAt_const.mul (ho p hp0 hp1)
change JetCalculus.mdir (1,0) jointSpace r (jointMean F x r)
(JetCalculus.materialIter (1,0) jointSpace r (jointMean F x r) (n+1) (jointMean F x r i)) (ρ,y)=_
rw [JetCalculus.mdir_congr_nonneg_time (1,0) jointSpace r (jointMean F x r)
(JetCalculus.materialIter_timeSmooth _ _ _ (jointMean_timeSmooth hF x hr.le hl)
(jointMean_timeSmooth hF x hr.le hl i) (n+1)) hs
(fun ht0 ht1 z => he hF x hr hlam hl ht0 ht1 i z) h0 h1 y,
JetCalculus.mdir_const_mul_at _ _ _ _ ((ho (ρ,y) (by dsimp; linarith) h1).differentiableAt (by simp)),
TensorSum.outer_material hF x hr hlam hl h0 h1 hAB]
end LogConcaveSampling
end
end
end
end OAI
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